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Amateur Quest For Lychrel Numbers

Posted by timothy on from the martin-gardner-would-be-proud dept.

Habberhead writes "Some people are aware of the quest for a palindromic solution for the number 196. Basically any number that doesn't form a palindrome by reversing and adding its digits is known as a Lychrel Number. (Sequence Number A023108 of Sloan's On-Line Encyclopedia of Integer Sequences) The number 196 happens to be the first of them. In over a year's worth of time, and more than 2 quadrillion calculations, this guy at www.p196.org has reversed and added the number over 100 MILLION times. His current answer is over 41 million digits long! Apparently he and a few others are also working on a distributed computing program for finding larger and larger Lychrel Numbers. It looks like they have in mind a Seti@Home style program with visible results."

9 of 310 comments (clear)

  1. Simple Example by teetam · · Score: 5, Interesting
    Consider 196:
    196+691 = 887 (which is not a palindrome)
    Apply the same for 887, 887+788 = 1675 (not a palindrome)

    Apparently, you can go on forever like this without ever reaching a palindrome!

    152, on the other hand, which I picked randomly, quickly reaches 707 which is a palindrome.

    Personally, I don't find this interesting at all. I posted a story a week ago about the prime number problem being solved for the first time with a deterministic algorithm and it was rejected by /. OOPS! Did I just go offtopic? Sorry, mods!!!

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  2. Re:what? by cperciva · · Score: 5, Informative

    256 + 652 = 908
    908 + 809 = 1717
    1717 + 7171 = 8888, which is a palindrome.

    However,
    196 + 691 = 887
    887 + 788 = 1675
    1675 + 5761 = 7436
    7436 + 6347 = 13783
    and contining on for a few million digits still doesn't end up at a palindrome.

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  3. Real world applications? by MattC413 · · Score: 4, Interesting

    What are some real-world applications that this process generates?

    Maybe some psuedo-random number generation with the huge strings of numbers that this comes up with?

    Any way that this could be used in some sort of encryption?

    There HAS to be some useful purpose to this.. There must be, or it wouldn't be the way it is! *twitch, twitch*

    -Matt

  4. Arbitrary definition of a palindrome? by PseudoThink · · Score: 4, Insightful

    Seems to me their palindrome test is a bit limited, since they only appear to be testing base-10 numbers. What's the use in that? Why not test base-2 or base-16 or whatever? Probably because there is no useful application to this arithmetic curiosity?

  5. Re:In a nutshell.... by Andrew+Allan · · Score: 5, Funny

    I've found another one!!!

    Try doing it with 691! ;-)

  6. Re:In a nutshell.... by tealover · · Score: 4, Insightful

    If you're going to copy stuff, you should at least give credit or show a link to the site that you're stealing from.

    This link comes from a link on the www.p196.org page.

    Moderators: Please mod the parent poster down for dishonesty.

    Thanks.

    --
    -- You see, there would be these conclusions that you could jump to
  7. Re:All the apathy here... by pomakis · · Score: 4, Insightful
    What would be interesting is coming up with a proof of why 196 exhibits this peculiar property. Until then, it's actually impossible to prove that a number is a Lychrel number. In fact, it's impossible to prove that there are any Lychrel numbers. Whose to say that the 20 billionth iteration of 196 isn't going to result in a palindrome?

    Until and unless there's a proof of why Lychrel numbers exist, the whole concept is quite uninteresting beyond a passing "neat".

  8. Re:All the apathy here... by ddstreet · · Score: 5, Insightful
    ALL numbers up to 10,000 become palindromes very quickly... except for the number 196?

    By definition the numbers 691, 887, 788, 1675, 5761, 7436, and 6347 must also have the same problem, since they're in the chain following 196.
    196 + 691 = 887
    887 + 788 = 1675
    1675 + 5761 = 7436
    7436 + 6347 = 13783

  9. A little logic by SiliconEntity · · Score: 5, Insightful

    I'm not familiar with this problem, so what I'm going to say is probably well known to students in the field.

    It seems like the best way to produce a palindrome on the next step is for the sum of the kth digit and the kth-from-the-end digit to be less than 10. Then there will be no carries and we get a nice palindrome.

    For random numbers, the chance of this being true is 1/2 for each digit in the first half of the number. Therefore with a number of length 2n digits, the chance that it will be palindromic on the next step is 1 in 2^n. (That' s one in two-to-the-nth power.)

    If a number is not a palindrome on one step, it will become about one digit longer from the reverse-and-add. So at each step that it is not palindromic, the chance that it ever will become palindromic decreases.

    From this perspective, it's not surprising that most small numbers become palindromes after a few steps, but that as we get to larger numbers we will find more and more that seem to never become palindromic. After some length the chance of ever again getting a palindrome is so remote that there is no point in continuing - your computer is more likely to make a mistake than for the number to happen to have the special form that can create a palindrome.

    196 just happens to be a number which "gets lucky", it escapes out of the small-number region where most form palindromes. Once you get past a dozen or so steps you'll probably never get a palindrome.

    There doesn't have to be anything special about 196, it's all a matter of chance and odds.

    That's how I see it, anyway.